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arXiv · 2006.13935

Primality of closed path polyominoes

Carmelo Cisto·Francesco Navarra
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Abstract

In this paper we introduce a new class of polyominoes, called closed paths, and we study the primality of their associated ideal. Inspired by an existing conjecture that characterizes the primality of a polyomino ideal by nonexistence of zig-zag walks, we classify all closed paths which do not contain zig-zag walks, and we give opportune toric representations of the associated ideals. To support the conjecture we prove that having no zig-zag walks is a necessary and sufficient condition for the primality of the associated ideal of a closed path. Finally, we present some classes of prime polyominoes viewed as generalizations of closed paths.

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Carmelo Cisto, Francesco Navarra. 2020-06-24. Primality of closed path polyominoes. https://doi.org/10.1142/s021949882350055x

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